Esakia space

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In mathematics, Esakia spaces are special ordered topological spaces introduced and studied by Leo Esakia in 1974.[1] Esakia spaces play a fundamental role in the study of Heyting algebras, primarily by virtue of the Esakia duality --- the dual equivalence between the category of Heyting algebras and the category of Esakia spaces.

Definition[edit]

For a partially ordered set (X,≤) and for x X, let x = {y X : yx} and let x = {y X : xy} . Also, for AX, let A = {y X : yx for some x A} and A = {y X : yx for some x A} .

An Esakia space is a Priestley space (X,τ,≤) such that for each clopen subset C of the topological space (X,τ), the set C is also clopen.

Equivalent definitions[edit]

There are several equivalent ways to define Esakia spaces.

Theorem:[2] The following conditions are equivalent:

(i) (X,τ,≤) is an Esakia space.
(ii) x is closed for each x X and C is clopen for each clopen CX.
(iii) x is closed for each x X and ↑cl(A) = cl(↑A) for each AX (where cl denotes the closure in X).
(iv) x is closed for each x X, the least closed set containing an up-set is an up-set, and the least up-set containing a closed set is closed.

Esakia morphisms[edit]

Let (X,≤) and (Y,≤) be partially ordered sets and let f : XY be an order-preserving map. The map f is a bounded morphism (also known as p-morphism) if for each x X and y Y, if f(x)≤ y, then there exists z X such that xz and f(z) = y.

Theorem:[3] The following conditions are equivalent:

(1) f is a bounded morphism.
(2) f(↑x) = ↑f(x) for each x X.
(3) f−1(↓y) = ↓f−1(y) for each y Y.

Let (X, τ, ≤) and (Y, τ′, ≤) be Esakia spaces and let f : XY be a map. The map f is called an Esakia morphism if f is a continuous bounded morphism.

Notes[edit]

  1. ^ Esakia (1974)
  2. ^ Esakia (1974), Esakia (1985).
  3. ^ Esakia (1974), Esakia (1985).

References[edit]

  • Esakia, L. (1974). Topological Kripke models. Soviet Math. Dokl., 15 147–151.
  • Esakia, L. (1985). Heyting Algebras I. Duality Theory (Russian). Metsniereba, Tbilisi.